American Institute of Mathematical Sciences

Advanced
July  2013, 12(4): 1745-1753. doi: 10.3934/cpaa.2013.12.1745

Remarks on nonlinear equations with measures

Moshe Marcus 1,

1. 

Department of Mathematics, Technion, Haifa 32000, Israel

Received  February 2011 Revised  July 2012 Published  November 2012

We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional assumptions on $g$, we present necessary and sufficient conditions for existence when $\mu$ is a finite measure. We also discuss the notion of solution when the measure $\mu$ is positive and blows up on a compact subset of $\Omega$.
Keywords: Orlicz - Sobolev spaces, absorption term, Semi-linear elliptic equation, weak solution, capacities..
Mathematics Subject Classification: Primary: 35J65, 46E30; Secondary: 46E35, 32U2.
Citation: Moshe Marcus. Remarks on nonlinear equations with measures. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1745-1753. doi: 10.3934/cpaa.2013.12.1745
References:
[1]

D. R. Adams and L. I. Hedberg, "Function Spaces and Potential Theory,'', Grundlehren Math. Wissen., 314 (1996).   Google Scholar

[2]

N. Aissaoui and A. Benkirane, Capacité dans les espaces d'Orlicz,, Ann. Sci. Math. Qu\'ebec, 12 (1994), 1.   Google Scholar

[3]

P. Baras and M. Pierre, Singularitès éliminables pour des équations semilinèaires,, Ann. Inst. Fourier, 34 (1984), 185.   Google Scholar

[4]

D. Bartolucci, F. Leoni, L. Orsina and A. Ponce, Semilinear equations with exponential nonlinearity and measure data,, Ann. I. H. Poincar\'e - AN, 22 (2005), 799.   Google Scholar

[5]

Ph. Benilan and H. Brezis, Nonlinear preoblems related to the Thomas-Fermi equation,, J. Evolution Eq., 3 (2003), 673.   Google Scholar

[6]

H. Brezis, Notes, unpublished (circa 1970)., (1970).   Google Scholar

[7]

H. Brezis H. and W. Strauss, Semilinear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.   Google Scholar

[8]

Th. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems,, J. Funct. An., 8 (1971), 52.   Google Scholar

[9]

M. A. Krasnoselskkii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,'', P. Noordhoff, (1961).   Google Scholar

[10]

J. B. Keller, On solutions of $\Delta u = f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503.   Google Scholar

[11]

M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case,, Arch. Rat. Mech. Anal., 144 (1998), 201.   Google Scholar

[12]

M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorption,, J. European Math. Soc., 6 (2004), 483.   Google Scholar

[13]

R. Osserman, On the inequality $\Delta u\geq f(u)$,, Pacific J. Math., 7 (1957), 1641.   Google Scholar

[14]

L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,'', Pitman Research Notes, 353 (1996).   Google Scholar

show all references

References:
[1]

D. R. Adams and L. I. Hedberg, "Function Spaces and Potential Theory,'', Grundlehren Math. Wissen., 314 (1996).   Google Scholar

[2]

N. Aissaoui and A. Benkirane, Capacité dans les espaces d'Orlicz,, Ann. Sci. Math. Qu\'ebec, 12 (1994), 1.   Google Scholar

[3]

P. Baras and M. Pierre, Singularitès éliminables pour des équations semilinèaires,, Ann. Inst. Fourier, 34 (1984), 185.   Google Scholar

[4]

D. Bartolucci, F. Leoni, L. Orsina and A. Ponce, Semilinear equations with exponential nonlinearity and measure data,, Ann. I. H. Poincar\'e - AN, 22 (2005), 799.   Google Scholar

[5]

Ph. Benilan and H. Brezis, Nonlinear preoblems related to the Thomas-Fermi equation,, J. Evolution Eq., 3 (2003), 673.   Google Scholar

[6]

H. Brezis, Notes, unpublished (circa 1970)., (1970).   Google Scholar

[7]

H. Brezis H. and W. Strauss, Semilinear second-order elliptic equations in $L^1$,, J. Math. Soc. Japan, 25 (1973), 565.   Google Scholar

[8]

Th. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems,, J. Funct. An., 8 (1971), 52.   Google Scholar

[9]

M. A. Krasnoselskkii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces,'', P. Noordhoff, (1961).   Google Scholar

[10]

J. B. Keller, On solutions of $\Delta u = f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503.   Google Scholar

[11]

M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case,, Arch. Rat. Mech. Anal., 144 (1998), 201.   Google Scholar

[12]

M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorption,, J. European Math. Soc., 6 (2004), 483.   Google Scholar

[13]

R. Osserman, On the inequality $\Delta u\geq f(u)$,, Pacific J. Math., 7 (1957), 1641.   Google Scholar

[14]

L. Véron, "Singularities of Solutions of Second Order Quasilinear Equations,'', Pitman Research Notes, 353 (1996).   Google Scholar

[1]

Jesus Idelfonso Díaz, Jean Michel Rakotoson. On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037
[2]

Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631
[3]

Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809
[4]

Guangyue Huang, Wenyi Chen. Uniqueness for the solution of semi-linear elliptic Neumann problems in $\mathbb R^3$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1269-1273. doi: 10.3934/cpaa.2008.7.1269
[5]

Út V. Lê. Contraction-Galerkin method for a semi-linear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 141-160. doi: 10.3934/cpaa.2010.9.141
[6]

Jason R. Morris. A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains. Conference Publications, 2009, 2009 (Special) : 574-582. doi: 10.3934/proc.2009.2009.574
[7]

Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481
[8]

Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761
[9]

Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047
[10]

Y. Kabeya, Eiji Yanagida, Shoji Yotsutani. Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems. Communications on Pure & Applied Analysis, 2002, 1 (1) : 85-102. doi: 10.3934/cpaa.2002.1.85
[11]

Houda Mokrani. Semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1619-1636. doi: 10.3934/cpaa.2009.8.1619
[12]

Vy Khoi Le. On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 809-818. doi: 10.3934/dcdss.2012.5.809
[13]

Duchao Liu, Beibei Wang, Peihao Zhao. On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1643-1659. doi: 10.3934/cpaa.2016018
[14]

Yongqin Liu. The point-wise estimates of solutions for semi-linear dissipative wave equation. Communications on Pure & Applied Analysis, 2013, 12 (1) : 237-252. doi: 10.3934/cpaa.2013.12.237
[15]

Jianhai Bao, Xing Huang, Chenggui Yuan. New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts. Communications on Pure & Applied Analysis, 2019, 18 (1) : 341-360. doi: 10.3934/cpaa.2019018
[16]

Ruirui Sun, Jinxia Li, Baode Li. Molecular characterization of anisotropic weak Musielak-Orlicz Hardy spaces and their applications. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2377-2395. doi: 10.3934/cpaa.2019107
[17]

Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029
[18]

Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255
[19]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565
[20]

Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, Aneta Wróblewska. Generalized Stokes system in Orlicz spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2125-2146. doi: 10.3934/dcds.2012.32.2125

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences